Search results for 'Number theory' (try it on Scholar)

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  1. Renling Jin (2000). Applications of Nonstandard Analysis in Additive Number Theory. Bulletin of Symbolic Logic 6 (3):331-341.score: 90.0
    This paper reports recent progress in applying nonstandard analysis to additive number theory, especially to problems involving upper Banach density.
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  2. Sue Ann Toledo (1975). Tableau Systems for First Order Number Theory and Certain Higher Order Theories. Springer-Verlag.score: 75.0
     
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  3. Erasmo Recami (1983). A Modified Large Number Theory Withconstant G. Foundations of Physics 13 (3):341-346.score: 60.0
    The inspiring “numerology” uncovered by Dirac, Eddington, Weyl,et al. can be explained and derived when it is slightly modified so to connect the “gravitational world” (cosmos) with the “strong world” (hadron), rather than with the electromagnetic one.The aim of this note is to show the following. In the present approach to the “Large Number Theory,” cosmos and hadrons are considered to be (finite)similar systems, so that the ratio ${{\bar R} \mathord{\left/ {\vphantom {{\bar R} {\bar r}}} \right. \kern-0em} {\bar (...)
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  4. Jeremy Avigad (2003). Number Theory and Elementary Arithmetic. Philosophia Mathematica 11 (3):257-284.score: 60.0
    is a fragment of first-order aritlimetic so weak that it cannot prove the totality of an iterated exponential fimction. Surprisingly, however, the theory is remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context.
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  5. J. Ellenberg & E. Sober (2011). Objective Probabilities in Number Theory. Philosophia Mathematica 19 (3):308-322.score: 60.0
    Philosophers have explored objective interpretations of probability mainly by considering empirical probability statements. Because of this focus, it is widely believed that the logical interpretation and the actual-frequency interpretation are unsatisfactory and the hypothetical-frequency interpretation is not much better. Probabilistic assertions in pure mathematics present a new challenge. Mathematicians prove theorems in number theory that assign probabilities. The most natural interpretation of these probabilities is that they describe actual frequencies in finite sets and limits of actual frequencies in (...)
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  6. J. Michael Dunn (1979). A Theorem in 3-Valued Model Theory with Connections to Number Theory, Type Theory, and Relevant Logic. Studia Logica 38 (2):149 - 169.score: 60.0
    Given classical (2 valued) structures and and a homomorphism h of onto , it is shown how to construct a (non-degenerate) 3-valued counterpart of . Classical sentences that are true in are non-false in . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (...)
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  7. Alessandro Berarducci & Benedetto Intrigila (1991). Combinatorial Principles in Elementary Number Theory. Annals of Pure and Applied Logic 55 (1):35-50.score: 57.0
    We prove that the theory IΔ0, extended by a weak version of the Δ0-Pigeonhole Principle, proves that every integer is the sum of four squares (Lagrange's theorem). Since the required weak version is derivable from the theory IΔ0 + ∀x (xlog(x) exists), our results give a positive answer to a question of Macintyre (1986). In the rest of the paper we consider the number-theoretical consequences of a new combinatorial principle, the ‘Δ0-Equipartition Principle’ (Δ0EQ). In particular we give (...)
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  8. Marcus Alfred, Petero Kwizera, James V. Lindesay & H. Pierre Noyes (2004). A Nonperturbative, Finite Particle Number Approach to Relativistic Scattering Theory. Foundations of Physics 34 (4):581-616.score: 54.0
    We present integral equations for the scattering amplitudes of three scalar particles, using the Faddeev channel decomposition, which can be readily extended to any finite number of particles of any helicity. The solution of these equations, which have been demonstrated to be calculable, provide a nonperturbative way of obtaining relativistic scattering amplitudes for any finite number of particles that are Lorentz invariant, unitary, cluster decomposable and reduce unambiguously in the nonrelativistic limit to the nonrelativistic Faddeev equations. The aim (...)
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  9. V. Tamma, C. O. Alley, W. P. Schleich & Y. H. Shih (2012). Prime Number Decomposition, the Hyperbolic Function and Multi-Path Michelson Interferometers. Foundations of Physics 42 (1):111-121.score: 54.0
    The phase φ of any wave is determined by the ratio x/λ consisting of the distance x propagated by the wave and its wavelength λ. Hence, the dependence of φ on λ constitutes an analogue system for the mathematical operation of division, that is to obtain the hyperbolic function f(ξ)≡1/ξ. We take advantage of this observation to decompose integers into primes and implement this approach towards factorization of numbers in a multi-path Michelson interferometer. This work is part of a larger (...)
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  10. Christopher Norris (2002). Putnam, Peano, and the Malin Génie: Could We Possibly Bewrong About Elementary Number-Theory? [REVIEW] Journal for General Philosophy of Science 33 (2):289-321.score: 48.0
    This article examines Hilary Putnam's work in the philosophy of mathematics and - more specifically - his arguments against mathematical realism or objectivism. These include a wide range of considerations, from Gödel's incompleteness-theorem and the limits of axiomatic set-theory as formalised in the Löwenheim-Skolem proof to Wittgenstein's sceptical thoughts about rule-following (along with Saul Kripke's ‘scepticalsolution’), Michael Dummett's anti-realist philosophy of mathematics, and certain problems – as Putnam sees them – with the conceptual foundations of Peano arithmetic. He also (...)
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  11. Toby Ord, Representations of Ω in Number Theory: Finitude Versus Parity.score: 48.0
    We present a new method for expressing Chaitin’s random real, Ω, through Diophantine equations. Where Chaitin’s method causes a particular quantity to express the bits of Ω by fluctuating between finite and infinite values, in our method this quantity is always finite and the bits of Ω are expressed in its fluctuations between odd and even values, allowing for some interesting developments. We then use exponential Diophantine equations to simplify this result and finally show how both methods can also be (...)
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  12. Mircea Radu (2013). Otto Hölder's 1892 “Review of Robert Graßmann's 1891 Theory of Number”. Introductory Note. Philosophia Scientiæ 17 (17-1):53-56.score: 48.0
    Hölder’s review of Robert Graßmann’s Theory of Number provides the first statement of Hölder’s most significant tenets concerning the distinction between the genetic and axiomatic presentation of mathematics, the mature expression of which is found in Hölder’s book The Mathematical Method of 1924. By translating Hölder’s review into English, I hope to make this unique document known to a wider public. In my introductory note I provide some context to Hölder’s paper and a few other remarks concerning the (...)
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  13. Rami Grossberg & Saharon Shelah (1986). On the Number of Nonisomorphic Models of an Infinitary Theory Which has the Infinitary Order Property. Part A. Journal of Symbolic Logic 51 (2):302-322.score: 48.0
    Let κ and λ be infinite cardinals such that κ ≤ λ (we have new information for the case when $\kappa ). Let T be a theory in L κ +, ω of cardinality at most κ, let φ(x̄, ȳ) ∈ L λ +, ω . Now define $\mu^\ast_\varphi (\lambda, T) = \operatorname{Min} \{\mu^\ast:$ If T satisfies $(\forall\mu \kappa)(\exists M_\chi \models T)(\exists \{a_i: i Our main concept in this paper is $\mu^\ast_\varphi (\lambda, \kappa) = \operatorname{Sup}\{\mu^\ast(\lambda, T): T$ is a (...)
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  14. Wolfram Pohlers (1998). Subsystems of Set Theory and Second Order Number Theory. In Samuel R. Buss (ed.), Handbook of Proof Theory. Elsevier. 137--209.score: 48.0
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  15. Manfred Szabo (1970). On the Original Gentzen Consistency Proof for Number Theory. In A. Kino, John Myhill & Richard Eugene Vesley (eds.), Intuitionism and Proof Theory. Amsterdam,North-Holland Pub. Co.. 409.score: 48.0
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  16. Steven C. Leth (1988). Some Nonstandard Methods in Combinatorial Number Theory. Studia Logica 47 (3):265 - 278.score: 46.0
    A combinatorial result about internal subsets of *N is proved using the Lebesgue Density Theorem. This result is then used to prove a standard theorem about difference sets of natural numbers which provides a partial answer to a question posed by Erdös and Graham.
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  17. Colin Mclarty (2010). What Does It Take to Prove Fermat's Last Theorem? Grothendieck and the Logic of Number Theory. Bulletin of Symbolic Logic 16 (3):359-377.score: 45.0
    This paper explores the set theoretic assumptions used in the current published proof of Fermat's Last Theorem, how these assumptions figure in the methods Wiles uses, and the currently known prospects for a proof using weaker assumptions.
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  18. T. M. Scanlon (1973). The Consistency of Number Theory Via Herbrand's Theorem. Journal of Symbolic Logic 38 (1):29-58.score: 45.0
  19. S. C. Kleene (1945). On the Interpretation of Intuitionistic Number Theory. Journal of Symbolic Logic 10 (4):109-124.score: 45.0
  20. Hilary Putnam (1960). An Unsolvable Problem in Number Theory. Journal of Symbolic Logic 25 (3):220-232.score: 45.0
  21. R. L. Goodstein (1947). Transfinite Ordinals in Recursive Number Theory. Journal of Symbolic Logic 12 (4):123-129.score: 45.0
  22. Yvon Gauthier (2008). From Fermat to Gauss: Indefinite Descent and Methods of Reduction in Number Theory Paolo Bussotti Augsburg, Erwin Rauner Verlag, 2006, 574 p. [REVIEW] Dialogue 47 (02):411-.score: 45.0
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  23. Yvon Gauthier (1978). Foundational Problems of Number Theory. Notre Dame Journal of Formal Logic 19 (1):92-100.score: 45.0
  24. B. Mazur (1994). Questions of Decidability and Undecidability in Number Theory. Journal of Symbolic Logic 59 (2):353-371.score: 45.0
  25. G. Kreisel (1952). On the Interpretation of Non-Finitist Proofs: Part II. Interpretation of Number Theory. Applications. Journal of Symbolic Logic 17 (1):43-58.score: 45.0
  26. W. D. Goldfarb & T. M. Scanlon (1974). The Ω-Consistency of Number Theory Via Herbrand's Theorem. Journal of Symbolic Logic 39 (4):678-692.score: 45.0
  27. W. Knorr (1976). Problems in the Interpretation of Greek Number Theory: Euclid and the 'Fundamental Theorem of Arithmetic'. Studies in History and Philosophy of Science Part A 7 (4):353-368.score: 45.0
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  28. Guillermo Restrepo & Leonardo Pachón (2007). Mathematical Aspects of the Periodic Law. Foundations of Chemistry 9 (2):189-214.score: 45.0
    We review different studies of the Periodic Law and the set of chemical elements from a mathematical point of view. This discussion covers the first attempts made in the 19th century up to the present day. Mathematics employed to study the periodic system includes number theory, information theory, order theory, set theory and topology. Each theory used shows that it is possible to provide the Periodic Law with a mathematical structure. We also show that (...)
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  29. Michael Van Laanen (2004). Encounters with Infinity: A Metamathematical Dissertation. New Age Book.score: 45.0
    This thesis is presented in the hope that it will resonate with mathematicians and others who are interested in analysis concepts and pure number theory.
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  30. Robert R. Tompkins (1968). On Kleene's Recursive Realizability as an Interpretation for Intuitionistic Elementary Number Theory. Notre Dame Journal of Formal Logic 9 (4):289-293.score: 45.0
  31. F. G. Asenjo (1965). The Arithmetic of the Term-Relation Number Theory. Notre Dame Journal of Formal Logic 6 (3):223-228.score: 45.0
  32. Jeremy Avigad, Kevin Donnelly, David Gray & Adam Kramer, Number Theory.score: 45.0
    1.1 Some examples of rule induction on permutations . . . . . . . 6 1.2 Ways of making new permutations . . . . . . . . . . . . . . . 7 1.3 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Removing elements . . . . . . . . . . (...)
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  33. Dorothy Bollman (1967). Formal Nonassociative Number Theory. Notre Dame Journal of Formal Logic 8 (1-2):9-16.score: 45.0
  34. Alan C. Bowen (1989). Boethian Number Theory: A Translation of the de Institutione Arithmetica with Introduction and Notes. Ancient Philosophy 9 (1):137-143.score: 45.0
  35. Harry Gonshor (1980). Number Theory for the Ordinals with a New Definition for Multiplication. Notre Dame Journal of Formal Logic 21 (4):708-710.score: 45.0
  36. Steven Orey (1955). Formal Development of Ordinal Number Theory. Journal of Symbolic Logic 20 (1):95-104.score: 45.0
  37. Albert A. Mullin (1961). Correlative Remarks Concerning Elementary Number Theory, Groups and Mutant Sets. Notre Dame Journal of Formal Logic 2 (4):253-254.score: 45.0
  38. Albert A. Mullin (1967). On New Theorems for Elementary Number Theory. Notre Dame Journal of Formal Logic 8 (4):353-356.score: 45.0
  39. Nino B. Cocchiarella (1984). Formal Number Theory and Compatibility. Teaching Philosophy 7 (4):361-362.score: 45.0
  40. D. Bollman & M. Laplaza (1973). A Set-Theoretic Model for Nonassociative Number Theory. Notre Dame Journal of Formal Logic 14 (1):107-110.score: 45.0
  41. John Myhill (1952). A Derivation of Number Theory From Ancestral Theory. Journal of Symbolic Logic 17 (3):192-197.score: 45.0
  42. Ivor Bulmer-Thomas (1985). Boethian Number Theory Michael Masi: Boethian Number Theory: A Translation of the De Institutione Arithmetica (with Introduction and Notes). (Studies in Classical Antiquity, 6.) Pp. 198; 8 Figures with Mathematical Diagrams and Musical Notation in Text. Amsterdam: Editions Rodopi, 1983. Paper, Fl. 60. [REVIEW] The Classical Review 35 (01):86-87.score: 45.0
  43. Wilbur Knorr (1985). Boethius, Boethian Number Theory: A Translation of the “De Institutione Arithmetica,” Trans. Michael Masi. (Studies in Classical Antiquity, 6.) Amsterdam: Rodopi, 1983. Paper. Pp. 197; 6 Illustrations. $27.75. Distributed in the U.S.A. By Humanities Press, Atlantic Highlands, N.J. [REVIEW] Speculum 60 (4):946-948.score: 45.0
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  44. Mariko Yasugi (1967). Interpretations of Set Theory and Ordinal Number Theory. Journal of Symbolic Logic 32 (2):145-161.score: 45.0
  45. Rosina Albano- Zinco (1975). On Gurwitsch's Number Theory. Graduate Faculty Philosophy Journal 5 (1):109-112.score: 45.0
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  46. Martin Davis (1990). Gödel Kurt. Zur Intuitionistischen Arithmetik Und Zahlentheorie (1933e). A Reprint of 41811. Collected Works, Volume I, Publications 1929–1936, by Kurt Gödel, Edited by Feferman Solomon, Dawson John W. Jr., Kleene Stephen C., Moore Gregory H., Solovay Robert M., and van Heijenoort Jean, Clarendon Press, Oxford University Press, New York and Oxford 1986, Even Pp. 286–294. Gödel Kurt. On Intuitionistic Arithmetic and Number Theory (1933e). English Translation by Stefan Bauer-Mengelberg and Jean van ... [REVIEW] Journal of Symbolic Logic 55 (1):346-346.score: 45.0
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  47. Gert H. Muller (1966). Review: Georg Kreisel, Some Concepts Concerning Formal Systems of Number Theory. [REVIEW] Journal of Symbolic Logic 31 (1):128-128.score: 45.0
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  48. David Nelson (1948). Review: R. L. Goodstein, Transfinite Ordinals in Recursive Number Theory. [REVIEW] Journal of Symbolic Logic 13 (3):171-171.score: 45.0
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  49. Abraham Robinson (1971). Review: James Ax, Simon Kochen, Diophantine Problems Over Local Fields I; James Ax, Simon Kochen, Diophantine Problems Over Local Fields II. A Complete Set of Axioms for $P$-Adic Number Theory; James Ax, Simon Kochen, Diophantine Problems Over Local Field III. Decidable Fields. [REVIEW] Journal of Symbolic Logic 36 (4):683-684.score: 45.0
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  50. Julia Robinson (1972). Review: Martin Davis, Application of Recursive Function Theory to Number Theory. [REVIEW] Journal of Symbolic Logic 37 (3):602-602.score: 45.0
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